#8159125

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#45052	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(2,4)(7,10)(8,12)(14,16)(18,22)(19,25)(20,28)(30,32)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(3,9)(7,10)(11,15)(14,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(34,40)(35,44)(36,48)(42,46);(1,2,6,4)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066321);